Question

An electric dipole is placed along the x-axis at the origin $O$. A point $P$ is at a distance of $20cm$ from this origin such that $OP$ makes an angle $\pi /3$ with the x-axis. If the electric field at $P$ makes an angle $\theta$ with the x-axis, the value of $\theta$ would be :

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{3}+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$
• C. $\frac{2\pi }{3}$
• D. ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Answer 1

The correct option is B $\frac{\pi }{3}+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

The electric potential at P due to dipole is $V=k\frac{p\cos \varphi }{r^2}$ where p be the dipole moment and $k=\frac{1}{4\pi {ϵ}_0}$

Here, $E_r=-\frac{\partial V}{\partial r}=k\frac{2p\cos \varphi }{r^3}$ and

$E_{\varphi }=-\frac{1}{r}\frac{\partial V}{\partial \varphi }=k\frac{p\sin \varphi }{r^3}$

From figure, $\tan \alpha =\frac{E_{\varphi }}{E_r}=\frac{1}{2}\tan \varphi =\frac{1}{2}\tan \frac{\pi }{3}$ (as $\varphi =\pi /3$)

or $\alpha ={\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Thus, the angle $\theta =\varphi +\alpha =\frac{\pi }{3}+{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$